We may assume . (otherwise there's nothing to prove).
By the fundamental theorem of calculus we have . Hence we can't have or for all x. (since we also assumed )

But then there exist points such that f'(a)<1 and f'(b)>1 and the intermediate value theorem now gives that f'(x) reaches all points in (f'(a),f'(b)) and trivially we can find p,q in this interval such that pq= 1.

We may assume . (otherwise there's nothing to prove).
By the fundamental theorem of calculus we have . Hence we can't have or for all x. (since we also assumed )

But then there exist points such that f'(a)<1 and f'(b)>1 and the intermediate value theorem now gives that f'(x) reaches all points in (f'(a),f'(b)) and trivially we can find p,q in this interval such that pq= 1.

The fundumental theorem of calculus has the assumption that is continuous,

which is not given, and there are example of derivatives which are not continuous, so I think your argument is